1. Field of the Invention
The present invention generally relates to the shaping of deformable materials, and more particularly to a method of forming metal sheet into useful articles, wherein tooling is designed using mathematical models that rely on finite element analysis (FEA) techniques to optimize forming operations, tooling design and product performance in the formed articles.
2. Description of Related Art
Many articles are made by stamping, pressing or punching a base material so as to deform it into a piece or part having a useful shape and function. The present invention is concerned with mathematical modelling of the mechanics of such material flow and deformation, and is particularly concerned with the deformation of metal (e.g., aluminum) sheet using tools and dies, to produce a wide variety of products, from beverage cans to components for automotive applications.
When designing the shape of a product, such as a beverage can, it is important to understand how the deformation process will affect the blank of sheet metal. Finite element analysis codes, available from a variety of companies, can be used to analyze plasticity, flow and deformation to optimize forming operations, tooling design and product performance in product designs. These models may result in tooling which improves the quality of a product as well as reducing its cost. The predictive capability of such finite element models is determined to a large extent by the way in which the material behavior is described therein.
In order to appreciate the complexities of modelling the deformation process, it is helpful to understand some basic concepts of mechanical metallurgy, including the concepts of yield stress, workhardening, and strain path.
When some type of external loading device, such as a tensile test machine deforms a metal, the initial response is elastic, with a linear relationship between stress and strain. At some value of the stress, determined by the microstructure of the metal, plastic deformation begins and the response is non-linear, and comprises elastic plus plastic deformation. The yield stress defines the strength of the metal at the condition where plastic deformation is initiated. Deformation beyond the yield stress is characterized by workhardening which causes the stress to increase at an ever decreasing rate until a failure mechanism intervenes and the sample breaks. Thus, the yield stress value and the workhardening curve are the two fundamental entities that define the plastic deformation of metals.
The forming of metal sheet into industrial or consumer products (e.g., cans and automotive components) occurs under multi-axial straining conditions, not the simple uniaxial path described above (tensile testing). In such cases the deformation is described by the strain path. The strain path is defined by the plastic strain tensor.
A tensor is a mathematical entity that is useful in describing various physical properties. Most physical properties can be expressed as either a scalar, a vector, or a tensor. A scalar quantity is one which can be specified with a single number (e.g., temperature or mass), while a vector quantity is one which requires two values, such as magnitude and direction (e.g., velocity or force). A tensor quantity is a higher-order entity that requires more than two values, i.e., more than a single magnitude and direction. For example, a stress tensor is a 3xc3x973 array, each term of which is defined by the stress acting on a given plane, in a given direction. As two direction cosines are required for transformations, the stress tensor is a second order tensor.
The plastic strain (or strain rate) tensor is a second order tensor, which can be expressed as a 3xc3x973 matrix and, in principal axes, has the form:
Common strain paths and their associate values for the plastic strain tensor components are given below:
The concepts of the uniaxial stress-strain curve are extended to multi-axial plasticity by defining an effective stress and an effective strain, "sgr"eff and xcex5eff, which are functions of the components of the stress and plastic strain tensors. The concepts of the yield stress and workhardening are then extended to multi-axial conditions through the use of "sgr"eff and xcex5eff in place of the "sgr" and xcex5 of the uniaxial case. Specifically, the effective stress is given by the second invariant of the stress tensor, and plasticity is referred to as either J2 or von Mises.
For an isotropic sheet of metal, the plasticity properties do not depend on direction or strain path, and the uniaxial stress-strain curve is all that is required to characterize the forming of sheet into a product. When aluminum sheet is rolled, however, it is anisotropic, meaning that some of the mechanical properties will not be the same in all directions. Because rolled sheet is anisotropic, yield stress as well as workhardening depend on both direction in the sheet and strain path. For example, in aluminum can body stock, the stress-strain curve for a sample cut with its tensile axis in the rolling direction lies below that for a sample cut in the transverse direction. Under multi-axial stress conditions one must now replace the concept of a yield point with that of a yield surface which, in multi-dimensional stress space, defines the boundary between elastic and plastic response. Workhardening manifests itself as an increase in the distance from the origin of stress to a point on the yield surface. One must also allow for the possibility that the workhardening rate may depend on the strain path. Thus, workhardening changes not only the size of the yield surface, but also its shape.
The anisotropy of sheet is determined by crystallographic texture, that is, by the orientations of the crystals that make up the sheet. As single crystal properties are highly anisotropic, the anisotropy of sheet depends on the distribution of orientations of the crystals that comprise it. Thus the orientation distribution function (ODF) is a fundamental property of sheet. There are various types of analysis programs that use crystallographic texture.
The crystallographic texture of sheet, in the form of pole figures, is obtained experimentally using X-ray or neutron diffraction. The ODF and the weights table are calculated from the pole figure data. The latter is particularly important as it defines the volume fraction of crystals having a particular orientation. Typically, the weights for at least 600 discrete orientations are determined by analysis of experimental diffraction data and provide the crucial input for crystal plasticity calculations.
Crystal plasticity theory allows a stress-strain response for a material to be calculated using a given crystallographic texture and specified strain path. A material point simulator (MPS) is an analysis technique that incorporates crystal plasticity theory. Under crystal plasticity theory, the response of a small amount of material subject to a specified strain path is calculated. The response of the aggregate is calculated from the weighted responses to each of the crystals contained in it. Single crystal yield stress and workhardening parameters are determined by an iterative procedure to match prediction from the simulation to a measured stress-strain curve (generally uniaxial tension or compression). Having so determined the single ctystal properties, the stress strain behavior for any desired strain path can be calculated. In addition to conventional workhardening, the calculationsusually include the evolution of texture during deformation along the strain path. In fact, comparison of measured and predicted textures after deformation provides the principal means of validation of material point simuators.
A further analysis technique that is used to model the forming and performance of products from sheet is finite element analysis (FEA). An FEA subdivides the sheet into a number of elements, typically from a few hundred for a simple analysis to 100,000 or more for complex parts and forming processes. The tooling used to form a part is also meshed, and contact between the tooling and sheet is allowed so that the simulated motion of the tooling in the model deforms the sheet and makes a virtual part just as real tooling makes a part in a plant. Thus it is not necessary to know a priori the strain path followed by each element during the forming operation; it is simply a response to the motion of the tooling. Examples of the use of FEA for forming products from sheet are given in U.S. Pat. Nos. 5,128,877, 5,379,227, 5,390,127, and 5,402,366. The first three of these patents disclose methods for aiding sheet metal forming tools, which include representing the sheet metal as a mesh and including a plurality of nodes and associated elements. A computer determines the stress state of a sampling point based on an incremental deformation theory of plasticity (the described xe2x80x9cdisplacement methodxe2x80x9d is an FEA). The fourth patent discloses a method for simulating a forming operation using FEA and a particle flow model. These approaches do not involve any distinction between the anisotropic and isotropic characteristics of the sheet.
The plasticity properties of an individual element (or, more precisely, at each integration point within an element) are defined by the definitions of the yield surface and the hardening law, which comprise the essential material definitions required for the analysis. The vast majority of FEA use isotropic, von Mises plasticity for the former and a simple uniaxial stress-strain curve for the latter. A difficulty often exists in that the strains for a forming operation may exceed (in some cases by a factor of 10 or more) those achieved in the laboratory characterization of the sheet. In such a case, the FE analyst must provide the code with an extrapolation of the experimental data to strains in excess of those imposed by the tooling during the forming operation. This requirement is not a trivial task, as the hardening depends on both strain and strain path, due to the evolution of texture during the forming operation.
In many cases and especially for aluminum alloy sheet, anisotropy should be included in an FEA. Two basic options exist to do so. Over the past 50 years, a variety of analytical functions have been proposed to replace that of the isotropic von Mises. Notable are formulations by Hill in 1948, 1979, and 1990, Karafillis and Boyce in 1993, and Barlat in 1989, 1991 and 1997. The analytical function approach suffers two difficulties. First, since the function is a relatively simple, closed-form, algebraic expression it can only provide an approximation to the shape of the actual yield surface in six-dimensional stress space. In fact, in many cases the allowable stress space for analytical yield functions has been reduced to those appropriate for plane stress deformation. Secondly, the constants in these functions must be determined experimentally, from laboratory measurements of the anisotropy of the yield stress and/or r-value (ratio of width to thickness strain in a tensile test) for various strain paths and directions in the sheet. Typically, five or more experimental measurements must be made in order to evaluate the constants of an analytical yield function.
The second option is to use crystal plasticity to define the properties of each element. In essence, this means running a material point simulator calculation for each integration point of each element at each iteration of the FEA. While use of analytical functions increase computer processing (CPU) time by about a factor of two or three compared to a von Mises calculation, use of fully-coupled crystal plasticity can increase CPU time by orders of magnitude, and currently is feasible for only the smallest of models and is not practical for simulation of any real forming operation. It would, therefore, be desirable to provide a method of including consideration of anisotropy in an FEA without paying the enormous cost of requiring a full crystal plasticity calculation for every iteration (or even every tenth or hundredth iteration) of an analysis. It would be further advantageous if the method could include a characterization of yield surface and hardening that was defined in a six-dimensional stress space simulated by the FEA.
It is therefore one object of the present invention to provide an improved method of forming sheet metal into various articles.
It is another object of the present invention to provide such a method which uses mathematical (computational) models to optimize tooling designs and forming operations to give desired properties in the formed articles.
It is yet another object of the present invention to provide such a mathematical model that relies on finite element analysis (FEA) techniques, and takes into account anisotropic properties of the metal sheet without requiring excessive computational time.
It is the principal object of the present invention to provide a method to incorporate anisotropy in to a finite element analysis without the normal penalty in CPU time for doing so by decoupling the anisotropy calculations from the finite element calculations.
The foregoing objects are achieved by the following four steps in an analysis of a forming operation:
1. Uniaxial tension (or compression) curves and crystallographic texture data obtained experimentally from the sheet are used to calibrate the constants in an appropriate crystal plasticity material point simulator. The material point simulator can then be used to generate effective stressxe2x80x94effective strain curves for a variety of possible strain paths. These will form a set with an upper and lower bound.
2. The finite element analysis is preferably done using a local coordinate system that follows the rigid-body motion of the sheet during forming. In this way, the plastic strain (or strain rate) tensor will always be defined in a coordinate system consisting of directions parallel to the rolling direction, perpendicular to the rolling direction and through the thickness of the sheet. Steps one and two define the anisotropy needed for the FEA.
3. The strain path must be determined for each finite element at every converged step (or at predetermined intervals in the analysis). This can be achieved in a variety of ways, with increasing complexity:
by inspection of the geometry of the tooling and forming operation, (e.g., rolling or ironing operations) or
by performing an isotropic analysis using a single stress-strain curve (say uniaxial tension) of the forming operation and extracting the required strain path in a post-processing mode for each element, or by calculating at each converged step of analysis a parameter that depends on the particular state of the strain tensor for each element.
4. An appropriate stress strain curve for each element is then selected from the family of curves described in (1) above. In the simplest case, the lower bound is chosen for all elements (independent of their actual paths). This gives a lower bound analysis with the lower limits for stresses in the sheet and tooling loads. The next level of sophistication is to define groups of elements having like strain paths (e.g., a set comprising the dome of a bulge) and assign one of the stress-strain curves from the set described in (1) above to each group. The procedure is seen as analogous to defining temperature-dependent stress strain curves in a finite element model, with the parameter defining the strain path taking the place of temperature.
The result is a finite element model that is in close agreement with experimentally-generated data, and one that requires much less computational (CPU) time compared to prior art methods.
The above as well as additional objectives, features, and advantages of the present invention will become apparent in the following detailed written description.